Abstract

The nonlocal boundary value problem for the parabolic differential equation in an arbitrary Banach space with the dependent linear positive operator is investigated. The well-posedness of this problem is established in Banach spaces of all -valued continuous functions on satisfying a Hölder condition with a weight . New Schauder type exact estimates in Hölder norms for the solution of two nonlocal boundary value problems for parabolic equations with dependent coefficients are established.

1. Introduction: A Cauchy Problem

It is known that (see, e.g., [1–5] and the references given therein) several applied problems in fluid mechanics, physics, and mathematical biology were formulated into nonlocal mathematical models. In general, such nonlocal problems were not well studied.

Before going to discuss the well-posedness of nonlocal boundary value problem, we will give the definition of positive operators in a Banach space and introduce the fractional spaces generated by positive operators that will be needed in the sequel.

Let be a Banach space and let be a linear unbounded operator densely defined in . The operator is said to be positive in the Banach space if its spectrum lies in the interior of the sector of angle , , symmetric with respect to the real axis, and if, on the edges of this sector, and , and outside of the sector the resolvent is subject to the bound
The infimum of all such angles is called the spectral angle of the positive operator and is denoted by . We call strongly positive in the Banach space if its spectral angle .

For positive operator in the Banach space , let us introduce the fractional spaces consisting of those for which the norm
is finite.

In the paper [6], the well-posedness in spaces of smooth functions of the nonlocal boundary value problem
for the differential equation in an arbitrary Banach space with the strongly positive operator was established. The importance of coercive inequalities (well-posedness) is well known [7, 8].

Noted that theory and methods for approximate solutions of local and nonlocal boundary value problems for evolution differential equations have been studied extensively by many researchers (see [9–39] and the references therein).

Before going to establish theorems on the well-posedness of nonlocal boundary value problem for parabolic equations in an arbitrary Banach space with the strongly positive dependent space operators, let us consider the abstract Cauchy problem for the differential equation
in an arbitrary Banach space with the strongly positive operators in with domain , independent of and dense in .

A function is called a solution of the problem (4) if the following conditions are satisfied.(i) is continuously differentiable on the segment??. The derivative at the endpoints of the segment is understood as the appropriate unilateral derivatives.(ii)The element belongs to for all , and the function is continuous on .(iii) satisfies the equation and the initial condition (4).

A solution of problem (4) defined in this manner will from now on be referred to as a solution of problem (4) in the space . Here, stands for the Banach space of all continuous functions defined on with values in equipped with the norm
From the existence of the such solutions, it evidently follows that and .

We say that the problem (4) is well posed in if the following conditions are satisfied.(1)Problem (4) is uniquely solvable for any and any . This means that an additive and homogeneous operator is defined which acts from to and gives the solution of problem (4) in .(2), regarded as an operator from to , is continuous. Here, is understood as the normed space of the pairs , , and , equipped with the norm
By Banach’s theorem, in and these properties, one has coercive inequality
where does not depend on??and .

Inequality (7) is called the coercivity inequality in for (4). If , then the coercivity inequality implies the analyticity of the semigroup , that is, the following estimates
hold for some . Thus, the analyticity of the semigroup is??necessary for the well-posedness of problem (4) in . Unfortunately, the analyticity of the semigroup is not sufficient for the well-posedness of problem (4) in .

Suppose that, for each , the operator generates an analytic semigroup with exponentially decreasing norm, when , that is, the following estimates
hold for some ,??. From this inequality, it follows that the operator exists and is bounded, and hence is closed in .

Suppose that the operator is Hölder continuous in in the uniform operator topology for each fixed , that is,
where and are positive constants independent of , and for .

An operator-valued function , defined and strongly continuous jointly in and for , is called a fundamental solution of (4) ifthe operator is strongly continuous in and for ,the following identity holds:
the operator maps the region into itself; the operator is bounded and strongly continuous in and for ,on the region , the operator is strongly differentiable relative to and , while
is also called development family, evolution operator, Green’s function, and so forth [14].

If the function is not only continuous, but also continuously differentiable on , and , it is easy to show that the formula
gives a solution of problem (4).

Now, we will give lemmas and estimates from [14] concerning the semigroup and the fundamental solution of (4) and theorem on well-posedness of (4) which will be useful in the sequel.

Lemma 1. For any ,? , and , one has the inequality
where does not depend on , and .

Lemma 2. For any , ? , ? and , the following estimates hold:
where and do not depend on , and .

Lemma 3. For any and , the following identities hold:

Lemma 4. For any , the following estimates hold:
where does not depend on and .

A function is said to be a solution of problem (4) in if it is a solution of this problem in and the functions?? and belong to .

As in the case of the space , we say that the problem (4) is well posed in if the following two conditions are satisfied.(1)For any and , there exists the unique solution in of problem (4). This means that an additive and homogeneous operator is defined which acts from to and gives the solution of (4) in .(2), regarded as an operator from to , is continuous. Here, is understood as the normed space of the pairs (,, and , equipped with the norm

We set equal to space, obtained by completion of the set of all smooth -valued functions on with respect to the norm
Let us give the following theorems on well-posedness of (4) in from [14]. To these, there correspond the spaces of traces , which consist of elements for which the norm
is finite.

Theorem 5. Suppose ,??. Suppose that the assumptions (9) and (10) hold and . Then, for the solution in of the Cauchy problem (4), the coercive inequality
holds, where does not depend on , and .

Theorem 6. Suppose ,??. Suppose that the assumptions (9) and (10) hold and . Then, for the solution in of the Cauchy problem (4), the coercive inequality
holds, where does not depend on , and .

Note that the spaces of smooth functions , in which coercive solvability has been established, depend on the parameters , and . However, the constants in the coercive inequalities depend only on . Hence, we can choose the parameters and freely, which increases the number of functional spaces in which problem (4)??is well posed. In particular, Theorems 5 and 6 imply theorems on well-posedness of the nonlocal boundary value problem (4) in .

Finally, in the paper [40], the initial-value problem
for the fractional parabolic equation in an arbitrary Banach space with the strongly positive operators was investigated. Here, is standard Riemann-Liouville’s derivative of order . The well-posedness of problem (26) in spaces was established. New exact estimates in Hölder norms for the solution of initial-boundary value problems for fractional parabolic equations were obtained.

In the paper [41], the nonlocal boundary value problem for the parabolic differential equation
in an arbitrary Banach space with the strongly positive operators in with domain , independent of and dense in , was investigated. The well-posedness of problem (27) in spaces was established. New exact estimates in Hölder norms for the solution of three nonlocal boundary value problems for parabolic equations were obtained.

In the present paper, the well-posedness of problem (27) in spaces is established. New Schauder type exact estimates in Hölder norms for the solution of two nonlocal boundary value problems for parabolic equations with dependent coefficients are established.

The paper is organized as follows. Section 1 is introduction. In Section 2, new theorems on well-posedness of problem (27) in spaces are established. In Section 3, theorems on the coercive stability estimates for the solution of two nonlocal boundary value parabolic problems are obtained. Finally, Section 4 is conclusion.

Now, we will give lemmas on the fundamental solution of (4) from paper [41].

Lemma 7. Assume that for any . Then, for any and , the following identity holds

Lemma 8. Under the assumption of Lemma 7 there exists the inverse of the operator in and the following estimate holds:

A function is called a solution of the problem (27) if the following conditions are satisfied.(i) is continuously differentiable on the segment .(ii)The element belongs to for all , and the function is continuous on .(iii) satisfies the equation and the nonlocal boundary condition (27).

We say that the problem (27) is well posed in if the following conditions are satisfied.(1)Problem (27) is uniquely solvable for any and any . This means that an additive and homogeneous operator is does not depend on defined which acts from to and gives the solution of problem (4) in .(2), regarded as an operator from to , is continuous. Here, is understood as the normed space of the pairs ,? , and , equipped with the norm
By Banach’s theorem in and these properties, one has coercive inequality
where does not depend on ??and .

Inequality (32) is called the coercivity inequality in for (27). ?If , then the coercivity inequality implies the analyticity of the semigroup . Thus, the analyticity of the semigroup is necessary for the well-posedness of problem (27) in . Unfortunately, the analyticity of the semigroup is not sufficient for the well-posedness of problem (27) in .

A function is said to be a solution of problem (27) in if it is a solution of this problem in and the functions?? and belong to .

As in the case of the space , we say that the problem (27) is well posed in if the following two conditions are satisfied.(1)For any and , there exists the unique solution in of problem (27). This means that an additive and homogeneous operator is defined which acts from to and gives the solution of (27) in .(2), regarded as an operator from to , is continuous. Here, is understood as the normed space of the pairs (,, and , equipped with the norm
If is a solution in of problem (27), then it is a solution in of this problem. Hence, by (13), we get the following representation for the solution of problem (27):
Using (27) and formula (34), we get
where
The main result of the present paper is the following theorem on well-posedness of (27) in the spaces .

Theorem 9. Suppose . Suppose that the assumptions (9) and (10) hold and . If , then for the solution in of the nonlocal boundary value problem (27) the coercive inequality
holds, where does not depend on , and .

Proof. The proof of Theorem 9 is based on Theorem 5 and the following estimate:
Let us estimate for any in , separately. We start with . Applying the inequality (30), we get
Therefore, we will estimate
We have that
where
By the definition of spaces and using estimates (9), we obtain
for any . We will consider separately the cases and . If , then
If , then
Combining these estimates, we get
for any . Therefore,
By the definition of spaces and using estimates (9), (18), (19), and (28), we obtain
for any . We will consider separately the cases and . If , then
If , then
Combining these estimates, we get
for any . Therefore,
Applying the triangle inequality and estimates (47) and (52), we obtain
From this estimate and estimate (39), it follows the following estimate
Now, we estimate . Applying inequality (30), we get
Using estimates (9), (18), (19), and (28), we obtain
for all . We will prove that
for any . If , then
If , then
Combining these estimates, we get (57). Applying estimates (55), (56), and (57), we obtain
Applying inequality (30), we get
Finally, we estimate . Applying inequality (30), we get
Using estimates (9), (18), (19), and (28), we obtain
for all . Applying estimates (63) and (62), we get
Combining the estimates (54), (60), (61), and (64), we get estimate (38). Theorem 9 is proved.

Furthermore, the method of proof of Theorem 6 and scheme of proof of Theorem 9 enable us to establish the following theorem on well-posedness of (27) in spaces .

Theorem 10. Suppose . Suppose that the assumptions (9) and (10) hold and . If , then for the solution in of the nonlocal boundary value problem (27) the coercive inequality
holds, where does not depend on , and .

Note that the spaces of smooth functions , in which coercive solvability has been established, depend on the parameters , and . However, the constants in the coercive inequalities depend only on . Hence, we can choose the parameters and freely, which increases the number of functional spaces in which problem (27) is well posed. In particular, Theorems 9 and 10 imply theorems on well-posedness of the nonlocal boundary value problem (27) in which is established in the paper [41].

3. Applications

First, we consider the nonlocal boundary value problem for the parabolic equation
where ,??, and ?? are given sufficiently smooth functions and and they satisfy every compatibility conditions which guarantee that the problem (66) has a smooth solution . Here,?? is a sufficiently large number.

We introduce the Banach spaces of all continuous functions satisfying a Hölder condition for which the following norms are finite:
where is the space of the all continuous functions defined on with the usual norm
It is known that the differential expression
defines a strongly positive operator acting in with the domain and satisfying nonlocal boundary conditions for any .

Therefore, we can replace the nonlocal boundary value problem (66) by the abstract nonlocal boundary value problem (27) in a Banach space with a strongly positive operator defined by formula (69).

Theorem 11. For the solution of nonlocal boundary value problem (66), the following coercive inequality:
holds, where does not depend on , ,??,??.

The proof of Theorem 11 is based on abstract Theorem 9 and on the following theorem on the structure of fractional spaces .

Second, on the range , we consider the nonlocal boundary value problem for the 2th order multidimensional parabolic equation
where ??and ?, and ?? are given sufficiently smooth functions. Here,?? is a sufficiently large number.

Let us consider a differential operator with constant coefficients of the form
acting on functions defined on the entire space??. Here, is a vector with nonnegative integer components,??. If is an infinitely differentiable function that decays at infinity together with all its derivatives, then, by means of the Fourier transformation one establishes the equality
Here, the Fourier transform operator is defined by the rule
The function is called the symbol of the operator and is given by
We will assume that the symbol
of the differential operator of the form
acting on functions defined on the space , satisfies the inequalities
for and for any . Assume that the all compatibility conditions hold which guarantees that the problem (71) has a unique smooth solution . This allows us to reduce the nonlocal boundary value problem (71) by the abstract nonlocal boundary value problem (27) in a Banach space of all continuous bounded functions defined on satisfying a Hölder condition with the indicator with a strongly positive operator defined by (77).

Theorem 13. For the solution of the nonlocal boundary value problem (71), the following coercivity inequality:
holds, where does not depend on??,??,??, ??.

The proof of Theorem 13 is based on abstract Theorem 9, on the coercivity inequality for an elliptic operator in , and on the following theorem on the structure of fractional spaces .

4. Conclusion

In the present study, the well-posedness of the nonlocal boundary value parabolic problem with dependent coefficients in Hölder spaces with a weight is established. In practice, new Schauder type exact estimates in Hölder norms for the solution of two nonlocal boundary value problems for parabolic equations with dependent coefficients are obtained. Moreover, applying the result of the monograph [14], the high order of accuracy single-step difference schemes for the numerical solution of the nonlocal boundary value parabolic problem with dependent coefficients can be presented. Of course, the coercive stability estimates for the solution of these difference schemes have been established without any assumptions about the grid steps.

Conflict of Interests

The authors declare that they have no conflict of interests.

Acknowledgments

The authors would like to thank Professor Pavel E. Sobolevskii, for his helpful suggestions. They??would like also to thank the referees for their efforts to improve the paper.